Two-Variable Reaction-Diffusion Systems

In this section we study the onset of Turing instabilities in more detail for two-variable systems. We will focus on systems with Neumann boundary conditions, which are most relevant for experimental systems. 

The growth rates of the spatial modes cos(A x) are the roots of the characteristic polynomial 

Equation (10.26) is called the dispersion relation it relates the growth rates of the spatial modes to the parameter values of the system. 

A spatial Hopf bifurcation, commonly called a wave bifurcation, corresponds to a pair of purely imaginary eigenvalues for some 0 Ci = 0 and C2 0. According to the stability conditions (10.23), T 0, and therefore 

In other words, for a Turing instability to occur, the activator must diffuse slower than the inhibitor. This is known as the principle of short-range activation and long-range inhibition. It is also known as local autocatalysis with lateral inhibition or local auto-activation-lateral inhibition (LALI), see for example [332, 319], local self-activation and lateral inhibition [280], or self-enhancement and lateral inhibition (SELI) [315] and has been applied to mechanisms other than reaction-diffusion. 

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The Turing condition for two-variable DIRWs, (10.85), has the same form as the Turing condition for two-variable reaction-diffusion systems, (10.31). Consequently, the uniform steady state (10.70) of a DIRW undergoes a Turing bifurcation with critical wavenumber... 

Spatial Hopf bifurcations or wave bifurcations can never occur in two-variable reaction-diffusion systems, see Sect. 10.1.2. This is no longer the case for reaction-transport systems with inertia. As shown in Sects. 10.2.1 and 10.2.2, spatial Hopf bifurcations are in principle possible in two-variable hyperbolic reaction-diffusions... 

We impose the first inequality in (11.3), since a Turing instability can occur in a two-variable reaction-diffusion system with constant parameters only if > 1. The second inequality ensures the positivity of D (t). We assume that the system (11.1) possesses a uniform steady state, (PuU)-PvU)) = (Pu.Pv). with Ej(Pu,Pv) = E2G0U, Pv) = which fulfills the stability conditions (10.23), and U is an activator and V an inhibitor. 

This is the dispersion relation for a two-variable reaction-diffusion system with a step-function diffusivity for V. It is the analog of (10.26) for homogeneous reaction-diffusion systems and relates the growth rates X of spatial perturbations to the parameter values of the system. In contrast to the homogeneous case, the dispersion relation (11.47) is a complicated expression that cannot be solved analytically if D D. A diffusion-driven instability of the uniform steady state of the system occurs if the stability condition (10.23) is satisfied and (11.47) has solutions with a positive real part. 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

The model which my colleagues and I have used over the past several years in our studies of spiral waves is simple, two-variable reaction-diffusion model of the Fitzhugh-Nagumo type [ 13,19]. It is a mathematical caricature of what is thought to take place in many real excitable systems. The model has the virtue of providing particularly fast time-dependent numerical simulations of... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Pearson has analyzed the effect of an immobile species on the Turing instability in two-variable activator-inhibitor systems for more general conditions [346]. Consider the 2+1 species system described by the following reaction-diffusion equations ... 

In a recent numerical study (17), spontaneous chemomechanicaJ pulsations were achieved by coupling a mechanically responsive sphere of gel with a chemical reaction which cannot produce oscillations by itself. The toy model used in this study is based on a two variables model of a quadratic autocatalytic reaction described by the following reaction-diffusion system dufdt = -h V u dv/dt = (12/7)w i +... 

It is noteworthy that the model is not exceptional one. The identical patterns can be generated in many reaction-diffusion systems, provided that they have the similar qualitative properties to the presented model. Moreover, it is worth to stress that the model contains two variables only, and therefore, it is simple one. One can expect more rich patterns in systems with three, fom, and more variables. 

From the differences that arise between the dynamics of the BZ reaction in an active and a passive gel, one may conclude that the dynamics of the active gel-BZ system does not result from a simple forcing of the gel by the chemistry. Numerical results clearly show that our three-variable model (including volume fraction p) constitutes a dynamical system different from the two-variable BZ reaction-diffusion system in a passive sphere. 

Fig. 7. A perspective plot of the spatio-temporal variation of the variable u(x, t) as computed with the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6), and the model parameters e = 0.01, a = 0.2, uo = ui = —1.5. (a) Stationary two-front pattern (D = 0.05) (b) two in-phase periodically oscillating fronts D = 0.03) (c) bursting pattern (D = 0.055) (d) two out of phase periodically oscilating fronts ( > = 0.02).

The above approaches to tabulation, whilst mostly applied in the simulation of combustion problems, have a general foundation that would be relevant to many kinetic systems. However, a special class of tabulation methods has been developed for flame simulations. If a fast exothermic reaction takes place between two components (e.g. a fuel and an oxidiser) of a gaseous system, then flames are observed. In premixed flames the fuel and the oxidiser are premixed before combustion takes place, whilst in non-premixed (diffusion) flames, the fuel and the oxidiser diffuse into each other, and the flame occurs at the boundary or flame front. Premixed and non-premixed flames are two extreme cases, but in many practical flames, continuous states between these two extremes will exist. Flames can be classified as laminar or turbulent according to the characteristics of the flow. Flames are special types of reaction—diffusion systems, characterised by high spatial gradients in temperature and species concentrations, and consequently reaction rates will have a high spatial variability. 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

Spatiotemporal pattern formation at the electrode electrolyte interface is described by equations that belong in a wider sense to the class of reaction-diffusion (RD) systems. In this type of coupled partial differential equations, any sustained spatial structure comes about owing to the interplay of the homogeneous dynamics or reaction dynamics and spatial transport processes. Therefore, the evolution of each variable, such as the concentration of a reacting species, can be separated into two parts the reaction part , which depends only on the values of the other variables at one particular location, and another part accounting for transport processes that are induced by spatial variations in the variables. These latter processes constitute a spatial coupling among different locations. 

We consider a general two-variable model involving reaction and diffusion, The PDE system is given by the following pair of equations ... 

Vanag and Epstein have formulated a four-variable model to understand pattern formation in the BZ-AOT system [449,453]. Their model builds on the Oregonator, see Sect. 1.4.8. It assumes that the chemistry within the water core of the droplets is well described by the two-variable Oregonator rate equations (1.131). It further assumes that the species in the oil phase are inert, since they lack reaction partners, the key reactants all being confined to the aqueous core of the droplets. Consequently, only transfer reactions occur for the activator B1O2 and inhibitor Br2 in the oil phase. The rate terms for the two transfer reactions are added to the rate terms of the two-variable Oregonator model. The reaction-diffusion equations of the four-variable model of the BZ-AOT system are given in nondimensionalized form by... 

This experimental result was compared with a calculation based on the NFT mechanism R1-R6, yet further simplified to a two-variable system [6]. Deterministic reaction-diffusion equations were solved numerically as described in Sect. 5.1.4, and the value of the flow rate at equistability was... 

In this model, e is the ratio of the time scales associated with the reactive dynamics of the two variables u and v, while D and are their diffusion coefficients. The parameters a and p characterize the local reactive dynamics. The FHN model was originally constructed as a simple scheme for describing electrochemical wave propagation in excitable nerve or cardiac tissue. The variable u corresponds to the potential while v represents ion currents in the nerve tissue. It has since been used extensively as a generic model that describes so-called excitable behavior of chemically reacting systems. In fact, as we shall show later in this chapter, it is possible to write a chemical reaction scheme whose rate law is of FUN form. ... 

Consider the general two-variable system of reaction-diffusion equations ... 

Abstract. The urea-urease system is a pH dependent enzymatic reaction that was proposed as a convenient model to study pH oscillations in vitro here, in order to determine the best conditions for oscillations, a two-variable model is used in which acid and substrate, urea, are supplied at rates kh and ks from an external medium to an enzyme-containing compartment. Oscillations were observed between pH 4 and 8. Thus the reaction appears a good candidate for the observation of oscillations in experiments, providing the necessary condition that kh > ks is met. In order to match these conditions, we devised an experimental system where we can ensure the fast transport of acid to the encapsulated urease, compared to that of urea. In particular, by means of the droplet transfer method, we encapsulate the enzyme, together with a suitable pH indicator, in a l-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC) lipid membrane, where differential diffusion of H+ and urea is ensured by the different permeability (Pm) of membranes to the two species. Here we present preliminary tests for the stability of the enzymatic reaction in the presence of lipids and also the successful encapsulation of the enzyme into lipid vesicles. 

The organic solvent is the most important variable as it controls partition and diffusion of the reactants between the two immiscible phases, the reaction rate, solubility, and swelling of permeability of the growing polymer. The solvent should be of such composition so as to prevent precipitation of the polymer before a high molecular weight has been attained. The final polymer should not dissolve in the solvent. The type of solvent will influence the characteristics of the physical state of the final polymer. Solvents such as chlorinated or aromatic hydrocarbons make useful solvents in this system. 

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